Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
Related questions
Question
![**Surface Area Calculation Tutorial**
**Objective:**
Find the surface area of the composite shape. Round your answer to the nearest tenth.
**Diagram Overview:**
The image shows a composite shape composed of a cylinder with a cone on top. The dimensions of the shape are as follows:
- The height of the cone is 2 meters.
- The height of the cylinder is 13 meters.
- The radius of both the cylinder and the base of the cone is 1 meter.
**Given:**
- Height of cone (\(h_c\)): 2 meters
- Height of cylinder (\(h\)): 13 meters
- Radius of cylinder and cone base (\(r\)): 1 meter
**Calculation Steps:**
1. **Surface Area of the Cylinder:**
- Lateral surface area of the cylinder: \(2\pi rh\)
- Base surface area of the cylinder: \(\pi r^2\)
2. **Surface Area of the Cone:**
- To find the lateral surface area of the cone, first find the slant height (\(l\)):
\[
l = \sqrt{r^2 + h_c^2}
\]
\[
l = \sqrt{1^2 + 2^2} = \sqrt{1 + 4} = \sqrt{5} \approx 2.2 \text{ meters}
\]
- Lateral surface area of the cone: \(\pi rl\)
3. **Total Surface Area:**
- Combine the lateral surface areas of the cylinder and cone, and the base area of the cylinder.
**Detailed Calculation:**
1. **Cylinder:**
\[
\text{Lateral surface area of the cylinder} = 2\pi rh = 2\pi \times 1 \times 13 = 26\pi \approx 81.7 \text{ square meters}
\]
\[
\text{Base surface area of the cylinder} = \pi r^2 = \pi \times 1^2 = \pi \approx 3.1 \text{ square meters}
\]
2. **Cone:**
\[
\text{Lateral surface area of the cone} = \pi rl = \pi \times 1 \times 2.2 \approx 2.2\pi \approx](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F35133500-1226-4a9c-afb4-40ee5a709984%2Fb77121ba-72a7-4ae6-ace9-5c686d1486f4%2Fo7ddkvk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Surface Area Calculation Tutorial**
**Objective:**
Find the surface area of the composite shape. Round your answer to the nearest tenth.
**Diagram Overview:**
The image shows a composite shape composed of a cylinder with a cone on top. The dimensions of the shape are as follows:
- The height of the cone is 2 meters.
- The height of the cylinder is 13 meters.
- The radius of both the cylinder and the base of the cone is 1 meter.
**Given:**
- Height of cone (\(h_c\)): 2 meters
- Height of cylinder (\(h\)): 13 meters
- Radius of cylinder and cone base (\(r\)): 1 meter
**Calculation Steps:**
1. **Surface Area of the Cylinder:**
- Lateral surface area of the cylinder: \(2\pi rh\)
- Base surface area of the cylinder: \(\pi r^2\)
2. **Surface Area of the Cone:**
- To find the lateral surface area of the cone, first find the slant height (\(l\)):
\[
l = \sqrt{r^2 + h_c^2}
\]
\[
l = \sqrt{1^2 + 2^2} = \sqrt{1 + 4} = \sqrt{5} \approx 2.2 \text{ meters}
\]
- Lateral surface area of the cone: \(\pi rl\)
3. **Total Surface Area:**
- Combine the lateral surface areas of the cylinder and cone, and the base area of the cylinder.
**Detailed Calculation:**
1. **Cylinder:**
\[
\text{Lateral surface area of the cylinder} = 2\pi rh = 2\pi \times 1 \times 13 = 26\pi \approx 81.7 \text{ square meters}
\]
\[
\text{Base surface area of the cylinder} = \pi r^2 = \pi \times 1^2 = \pi \approx 3.1 \text{ square meters}
\]
2. **Cone:**
\[
\text{Lateral surface area of the cone} = \pi rl = \pi \times 1 \times 2.2 \approx 2.2\pi \approx
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